Interpretation of NDTM in the definition of NP

TL;DR

This paper reexamines the foundational definition of NP, tracing it back to Turing's work and clarifying the role of nondeterminism and oracle-based models to address ambiguities in the P versus NP problem.

Contribution

It clarifies the original interpretation of NP through NDTM and distinguishes between different models, highlighting the loss of nondeterminism in standard definitions.

Findings

NDTM originally related to oracle models in Turing's work.

The verifier-based definition of NP obscures nondeterminism.

Clarification of NP's roots may impact understanding of P vs NP.

Abstract

In this paper, we interpret NDTM (NonDeterministic Turing Machine) used to define NP by tracing to the source of NP. Originally NP was defined as the class of problems solvable in polynomial time by a NDTM in the theorem of Cook, where the NDTM was represented as Query Machine of essence Oracle. Later a model consisting of a guessing module and a checking module was proposed to replace the NDTM. This model of essence TM has a fundamental difference from the NDTM of essence Oracle, but people still use the term NDTM to designate this model, which leads to the disguised displacement of NDTM and produces out the verifier-based definition of NP as the class of problems verifiable in polynomial time by a TM (Turing Machine). This verifier-based one has been then accepted as the standard definition of NP where comes from the famous equivalence of the two definitions of NP. Since then the notion of nondeterminism is lost from NP, which causes ambiguities in understanding NP and then great difficulties in solving the P versus NP problem. Since NP is originally related with Oracle that comes from the work of Turing work about Computability, it seems quite necessary to trace back to the work of Turing and clarify further the issue about NP.